Modal Triplet Theory and History-Dependent Stochastic Processes: Fixed-State Indivisibility, Markov Order, and the Quantum Boundary

This paper gives a measure-theoretic account of stochastic processes obtained by projecting an upper Modal Triplet Theory evolution. Given a selected probability law on upper initial data, measurable upper dynamics, and an observable map, the projected histories carry a unique path-space law and admit regular conditional kernels. Projection can make the observed process non-Markovian even when the upper process is Markovian or deterministic. The paper separates failure of the observed Markov property, absence of finite Markov order, failure of a chosen classical composition law, and quantum CP-indivisibility. Every classical path law still factorizes sequentially through history-dependent kernels and becomes Markovian on a history state space. An exact binary example has infinite observed Markov order while its memory compresses to one posterior scalar. Exponential action weights define valid kernels when their action and reference measure are supplied, but rewriting a positive kernel in Gibbs form is parameterization rather than prediction. A commutative path-space algebra does not derive quantum observables, Born probabilities, process tensors, or instruments. The remaining MTT task is to select the upper probability source and physical intervention structure from the same geometry.